A multi-party contract signing (MPCS) protocol is used for a group of signers to sign a digital contract over a network. We analyse the protocols of Mauw, Radomirović and Torabi Dashti (MRT), using the finite-state model checker Mocha. Mocha allows for the specification of properties in alternating-time temporal logic (ATL) with game semantics, and the model checking problem for ATL requires the computation of winning strategies. This gives us an intuitive interpretation of the verification problem of crucial properties of MPCS protocols. MRT protocols can be generated from minimal message sequences, depending on the number of signers. We discover an attack on fairness in a published MRT protocol with three signers and a general attack on abuse-freeness for all MRT protocols. For both attacks, we present solutions. The abuse-freeness attack leads us to a revision of the methodology to construct an MRT protocol. Following this revised methodology, we design a number of MRT protocols using minimal message sequences for three and four signers, all of whom have been successfully model checked in Mocha.